Answer :
To find out which statement is true regarding the tire diameters and their circumferences, we use the given equation for the least-squares line:
[tex]\[
\hat{y} = 0.1 + 3.1x
\][/tex]
The values in the equation are:
- [tex]\(0.1\)[/tex] is the intercept.
- [tex]\(3.1\)[/tex] is the slope.
Let's calculate the circumference ([tex]\(\hat{y}\)[/tex]) for each tire diameter ([tex]\(x\)[/tex]):
1. For a 30-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 30 = 0.1 + 93 = 93.1
\][/tex]
- This matches statement (a).
2. For a 28-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 28 = 0.1 + 86.8 = 86.9
\][/tex]
- This does not match statement (b), which claims a circumference of 99.3 inches.
3. For a 32-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 32 = 0.1 + 99.2 = 99.3
\][/tex]
- This matches statement (c), but statement (c) claims a different diameter-circumference pair of 86.9 inches, so it's incorrect in its statement.
4. For a 35-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 35 = 0.1 + 108.5 = 108.6
\][/tex]
- This does not match statement (d), which claims a circumference of 124.1 inches.
Based on the calculations, the true statement is:
- a.) A tire with a 30-inch diameter at the tread will have a circumference of 93.1 inches.
[tex]\[
\hat{y} = 0.1 + 3.1x
\][/tex]
The values in the equation are:
- [tex]\(0.1\)[/tex] is the intercept.
- [tex]\(3.1\)[/tex] is the slope.
Let's calculate the circumference ([tex]\(\hat{y}\)[/tex]) for each tire diameter ([tex]\(x\)[/tex]):
1. For a 30-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 30 = 0.1 + 93 = 93.1
\][/tex]
- This matches statement (a).
2. For a 28-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 28 = 0.1 + 86.8 = 86.9
\][/tex]
- This does not match statement (b), which claims a circumference of 99.3 inches.
3. For a 32-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 32 = 0.1 + 99.2 = 99.3
\][/tex]
- This matches statement (c), but statement (c) claims a different diameter-circumference pair of 86.9 inches, so it's incorrect in its statement.
4. For a 35-inch diameter tire:
[tex]\[
\hat{y} = 0.1 + 3.1 \times 35 = 0.1 + 108.5 = 108.6
\][/tex]
- This does not match statement (d), which claims a circumference of 124.1 inches.
Based on the calculations, the true statement is:
- a.) A tire with a 30-inch diameter at the tread will have a circumference of 93.1 inches.
